Intercept plus a dummy for every sector

Six stocks, two in each of three sectors, had these returns yesterday (in %):

• Tech: 3 and 5
• Energy: $-1$ and 1
• Utilities: 2 and 2

An analyst regresses the returns on an intercept and a dummy for each sector, $r = \beta_0 + \beta_T d_T + \beta_E d_E + \beta_U d_U + \varepsilon$, so the design matrix $X$ is $6 \times 4$.

(a) Find the rank and the null space of $X$. Describe every coefficient vector that minimizes the squared error.

(b) Which of these does the data determine: $\beta_0$, $\beta_T - \beta_E$, $\beta_0 + \beta_U$, $\beta_T + \beta_E + \beta_U$? State a general rule in terms of the subspaces of $X$.

(c) Which subspace do the residuals lie in, and what is its dimension?